A Pinch of Topology

Topology -- an important area of mathematics that has applications from
subatomic physics to large-scale astronomy -- was born when
mathematicians realized that there is something behind the notion of
nearness that does not depend on measurements. This qualitative view
of nearness implies that elastic distortions, compressions or
expansions don't change the essence of a shape. By contrast, tearing
an object changes it drastically: points on opposite sides of the tear
end up apart, no matter how close together they were.
Puncturing -- that is, taking away a point -- is also an essential
change. So the rule about not tearing or puncturing the surface is a
topological one.

Topology deals with all sorts of surfaces (and also
curves). A subarea called differential
topology concentrates on surfaces that are smooth. Think
of a mesh on the surface, like the latitude-longitude grid on the
globe. Smoothness means that the curves of the mesh don't have
corners. This corresponds to the intuitive idea of a smooth surface,
without creases, kinks, corners, etc. At a crease (such as the
equator in the figure above) it's impossible to find a smooth mesh.

It is also impossible to find a smooth mesh around a pinch
point, like the point at the center of this surface:

(Click on the picture to rotate it. Check the
instructions if necessary.)

This surface is called the Whitney umbrella, in honor of
Hassler Whitney, one of the creators of differential topology. You
can obtain it by taking an X made of two horizontal lines, and
dragging it up while at the same time closing the angle between the
lines. The center of the X traces the vertical line of
self-intersections that ends at the pinch point, where the two lines
collapse into one.
1

Any point that cannot be included in a smooth mesh is called a
singular point. Singular points are not ``bad'' -- in fact they can
be quite interesting -- but our rules don't allow them because we want
our surface to remain smooth. Yorick's attempts to turn the sphere
inside out by just pushing the halves through each other fail because
he's creating singular points: a crease the first time around, and a pinch
point the second time.